What time of day Friday are you arriving at WDW? You can buy your AP, then call DDE and order the card, they will send a fax or email to your resort that you can use. But you must be arriving and call during DDE business hours...and I would definately try to call before 4pm to give them time to answer the phone (they are pretty swamped).
If you are arriving at Disney on a weekend, no, you will need to order your AP and have it sent quickly, then call DDE before you leave so you can get the email. AS others have said, you can pay additional shipping for rapid delivery.